The compactness of Moser-Trudinger functionals with conical metric in the unit ball of ${\mathbb{R}}^N$

Let $\mathbb{B}$ be the unit ball in ${\mathbb{R}}^N$, $W_0^{1,N}\left(\mathbb{B}\right)$ is a standard Sobolev space. Zhang proved the extremal function of the Moser-Trudinger inequality as follows, \begin{align*} \int_{ $\mathbb{B}$} h_ε(x) e^{ α_N \left( 1 + ε\right) |u_ε|^{ \frac{N}{N-1} } } dx, \quad u_ε \in W_0^{1,N} ( $\mathbb{B}$ ) \cap $\mathcal{S}$, \end{align*} where ${\alpha }_N={\omega }_N^{\frac{1}{N-1}}$, ${\omega }_N$ is the area of the unit sphere in ${\mathbb{R}}^N$(see \citep{26}) . In this paper, we consider the compactness of the sequence $\{u_{\epsilon }{\}}_{\epsilon }$ and prove that it has a subsequence converging to a function in $C^1\left(\bar{\mathbb{B}}\right)$.